by Alexander Diesl, Mathematics
Teaching Effectiveness Award Essay, 2003
What do students in second-year math courses find the most difficult? If you ask such a student, the response is likely to be “Proofs.” Although most students understand the idea that a mathematical proof is a rigorous justification of a mathematical fact, few would consider themselves to be good at writing proofs. Worse, most students consider proof writing to be a skill that cannot readily be learned. Some people are simply “good at proofs,” and some are not. This conception is a source of great frustration for students, who feel as though they simply cannot continue to understand and enjoy mathematics.
What the students do not understand is that the ability to write mathematical proofs is not a result of genius but rather of an understanding of the language of mathematics. Students think that they lack fundamental understanding when they in fact lack only the ability to translate their intuition into mathematically precise statements. This type of problem is not unique to mathematics. It can be likened, for example, to the problem of a student of philosophy who, although he understands his convictions, cannot effectively defend them in a written format. The essential problem in mathematics is that these difficulties in writing proofs are rarely recognized as problems concerning language and communication. The issue with helping students learn to write proofs is twofold. First, students must be made aware of the fact that mathematicians use a specific vocabulary and manner when expressing their ideas. Second, students must be provided with opportunities, outside of tests and homework, in which to practice communicating mathematics within this new framework.
The solution that I developed addresses both of these concerns. I begin by deconstructing the process a mathematician uses when writing a proof. I show the students a few carefully worked examples in which I make sure to explain my writing style and use of jargon. These examples give the student a model that he can later imitate when writing his own proofs. The second step involves the students writing their own proofs. When doing a proof exercise, the students work in small groups. They are encouraged to discuss the ideas and the intuition behind the proof, but each student writes the actual proof on her own. The students then trade their finished proofs with each other and critique the work of their fellow students. Several students also write proofs on the board for a class critique. Oftentimes, the students are much better at identifying weaknesses and language problems in their classmates’ proofs than in their own. This practice gained through writing and critiquing is vital to the student’s internalization of the mathematical language. The class then becomes a community of mathematicians; the students are active participants in the process of writing and verifying proofs. This allows them to begin to develop their own mathematical vocabulary.
This method has proven quite successful in my most recent class in linear algebra. On the first exam, which was largely computation-based, the average score in each section was roughly the same. The second exam, however, was much more proof-based, and students in my sections were able to score a full 10% higher on average than students in other sections. In addition, through working together and communicating as mathematicians, many of my students have commented that they have gained a new appreciation for the subject. I believe that this analysis highlights the importance of encouraging students to practice expressing themselves in the language of the discipline which they are studying.